On Lie Derivations, Generalized Lie Derivations and Lie Centralizers of Octonion Algebras

Minahal Arshad1, M. Mobeen Munir1
1Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, Pakistan.

Abstract

Let L be a unital ring with characteristic different from 2 and O(L) be an algebra of Octonion over L. In the present article, our attempt is to present the characterization as well as the matrix representation of some variants of derivations on O(L). The matrix representation of Lie derivation of O(L) and its decomposition in terms of Lie derivation and Jordan derivation of L and inner derivation of O is presented. The result about the decomposition of Lie centralizer of O in terms of Lie centralizer and Jordan centralizer of L is given. Moreover, the matrix representation of generalized Lie derivation (also known as D-Lie derivation) of O(L) is computed.

Keywords: The Octonion algebra, Lie Derivations, D-Lie Derivations, Lie Centralizer

1. Introduction

It is well known that derivation is a function of an algebra which generalizes certain features of the derivative operator. It gives an interesting insight to understand the structure and local properties of an algebra. The concept to study the structure of ring theory and derivations though established long back, but got stimulated after Posner in [1] described some important results on the derivations of prime rings. A fundamental problem in the theory of derivations is to determine all the derivations on an algebra. Over the years, many important variants of derivation have been presented. Among these variants, Lie derivation and its generic extension is currently more interesting and attracting. Lie centralizer is not so common but very meaningful variant. These variants are being widely discussed now a days.

Let O be a unital algebra and the center of O is denoted by Z(O). We denote the commutator (Lie product) and Jordan product of x1, x2 by [x1,x2]=x1x2x2x1 and x1x2=x1x2+x2x1 respectively for all x1,x2O. We say that a ring O is an F-algebra (F is a field) if O is an F-vector space equipped with a bilinear product. Let D:OO be an additive map. We say D is a derivation (respectively Jordan derivation) if D[x1,x2]=[D(x1),x2]+[x1,D(x2)] (respectively D(x2)=D(x)x+xD(x)), for all x,x1,x2O. For an element αO, the mapping Iα:OO given by Iα(x)=xααx for all xO is called an inner derivation of O induced by α. Let ξ:OO be an additive map. The map ξ is said to be right (left) centralizer if ξ(x1x2)=x1ξ(x2) (ξ(x1x2)=ξ(x1)x2) for all x1,x2O. We say that φ is an Jordan centralizer if φ(x1x2)=φ(x1)x2 for all x1,x2O. An additive map ξ:OO is called a Lie centralizer if ξ[x1,x2]=[ξ(x1),x2] for all x1,x2O.

The characterization of Lie centralizer of quaternion algebra is given in [2]. The characterization of Lie centralizers and their generalizations is now widely studied on different kinds of algebras by many authors in ([3,4,5,6,7]). Fosner et al. characterised Lie centralizers of triangular tings and nest algebras in [3]. Ashrafi et al. computed multiplicative generalized Lie n-derivations of unital rings with idempotents and σ-centralizers generalized matrix algebras in [4] and [5] respectively. Fadaee et al. characterized Lie centralizers at the zero product of generalized matrix algebras in [6] and Lie triple centralizers of generalized matrix algebras in [7]. In [8], Martindale described the standard form of Lie derivation on certain primitive rings. Similar results have been discussed on von Neumann algebras by Miers [9]. Mokhtari et al. computed Lie derivations on trivial extension algebras in [10]. It is obvious that every Lie derivation is a generalized Lie derivation. Besides there are two different definitions of generalized Lie derivations in literature. One is introduced by Atsushi Nakajima [11] which is stated as: for an additive map F:OO, we say that F is generalized Lie derivation (also known as D-Lie derivation) if there exist a Lie derivation D:OO such that F[x1,x2]=[F(x1),x2]+[x1,D(x2)] for all x1,x2D and the other is by Bojan Hvala [12] which is stated as: for an additive map F:OO, we say that F is generalized Lie derivation if there exist a linear map D:OO such that F[x1,x2]=F(x1)x2F(x2)x1+x1D(x2)x2D(x1) for all x1,x2D. Both definitions are equally being discussed by mathematicians. Hvala discussed the generalized Lie derivations on rings and proved that every generalized Lie derivation on a prime ring can be written as the sum of a generalized derivation and a central map. Benkovič [13] proved that every generalized Lie derivation from a unital algebra onto a unitary bimodule can be written in the sum of a generalized derivation and a central map that vanishes on the commutators of the algebra. The description of generalized Lie derivation of Lie ideals of prime algebras and nonlinear generalized Lie derivation of some classical triangular algebras are respectively given in [14, 15]. The generalized Lie derivations of prime rings are discussed in [16] by using both definitions. Hvala’s definition of generalized Lie derivation covers both generalized derivations and D-Lie derivations. On the other hand, Nakajima’s definition is more favourable which unifies the notions of Lie derivation and Lie centralizer. We will, in particular focus on Nakajima’s definition to compute the matrix representation of generalized Lie derivations of algebra of Octonion. The discussion about local and 2-local derivation of Octonion algebra have been discussed in [17]. Later, this discussion was extended to Cayley algebras in [18].

Mainly, our focus in this article is to describe the matrix representation as well as the characterization of Lie derivation of Octonion algebras equipped with commutator product. Authors characterized in [19], the Lie triple derivations of algebra of tensor product of some algebra T and quaternion algebra. Ghahramani et al. in [20] proved results on the characterization of generalized derivation and generalized Jordan derivation of ring of quaternion and in [21] discussed the characterization of Lie derivation and its natural generic extension of quaternion ring.

This article is arranged in the following order: Section 2 contains some minor details of Octonion algebras equipped with commutator product denoted by O. In Sections 3 and 4, matrix representation of Lie derivation as well as decomposition of Lie derivation of octonion algebra in terms of Lie derivation and Jordan derivation of L and inner derivation of O is presented. Section 5 contains the characterization of Lie centralizer of Octonion algebras. In Section 6, the matrix representation of generalized Lie derivation is computed.

2. The Octonion Algebra O

Let L be an arbitrary 2-torsion free unital ring. The octonion algebra (denoted by O) over L is a class of non-associative algebra. It is a unital nonassociative algebra of dimension 8 with the basis B={e0,e1,e2,e3,e4,e5,e6,e7} and the product defined in the following table.

ej
ei ei.ej e0 e1 e2 e3 e4 e5 e6 e7
e0 e0 e1 e2 e3 e4 e5 e6 e7
e1 e1 e0 e3 e2 e5 e4 e7 e6
e2 e2 e3 e0 e1 e6 e7 e4 e5
e3 e3 e2 e1 e0 e7 e6 e5 e4
e4 e4 e5 e6 e7 e0 e1 e2 e3
e5 e5 e4 e7 e6 e1 e0 e3 e2
e6 e6 e7 e4 e5 e2 e3 e0 e1
e7 e7 e6 e5 e4 e3 e2 e1 e0

<span id="corank1" label="corank1"

The table can be summarized as follows: ei.ej={ej,if i=0;ei,if j=0;δije0+ϵijkek,otherwise, where δij is the Kronecker delta and ϵijk is a completely antisymmetric tensor with value +1 when ijk=123, 145, 176, 246, 257, 347, 365.
An octonion x is of the form x=x0e0+x1e1+x2e2+x3e3+x4e4+x5e5+x6e6+x7e7 with real coefficients xi. By using the product defined in the table given above, we can have the following relations; [e1,e2]=2e3,[e1,e3]=2e2,[e1,e4]=2e5,[e1,e5]=2e4,[e1,e6]=2e7,[e1,e7]=2e6,[e2,e3]=2e1,[e2,e4]=2e6,[e2,e5]=2e7,[e2,e6]=2e4,[e2,e7]=2e5,[e3,e4]=2e7,[e3,e5]=2e6,[e3,e6]=2e5,[e3,e7]=2e4,[e4,e5]=2e1,[e4,e6]=2e2,[e4,e7]=2e3,[e5,e6]=2e3,[e5,e7]=2e2,[e6,e7]=2e1. Using the above product on the basis as Lie product and extend it by linearity, we can equip this product on O.

3. Lie Derivation of Octonion Algebra O

In this section, we compute matrix representation of Lie derivation of the octonion Algebra. Let D:OO be a Lie derivation. D admits a matrix representation with respect to the basis, which is an 8×8 matrix [D]=(βij)T whose entries are defined by (1)D(ei1)=j=18βijej1,1i8. Each column of [D] is an element of O.

Theorem 1. The algebra of Lie derivations of Octonions is generated by the following matrices: =(β11000000000β23β24β25β26β27β280β230β34β35β36β37β380β24β340β27+β36β28β35β25β38β26+β370β25β35β27β360β56β57β580β26β36β28+β35β560β23β58β24+β570β27β37β25+β38β57β23+β580β34β560β28β38β26β37β58β24β57β34+β560)

Proof. Let D be a Lie derivation of O. Then we write D(ei1)=j=18βijej1,1i8 for some arbitrary βijsL. Applying D on the identity [e0,ej]=0 for 1j7. So, we get β1j=0 for 2j7.
Applying D on identities e3=12[e1,e2], e3=12[e4,e7] and e3=12[e6,e5], we get D(e3)=12(2β22e32β24e12β25e62β26e7+2β27e4+2β28e5+2β33e32β34e2+2β35e52β36e42β37e7+2β38e6)=(2β52e62β53e52β54e4+2β55e3+2β56e22β57e12β82e52β83e62β84e7+2β86e1+2β87e2+2β88e3)=(2β62e72β63e4+2β64e5+2β65e22β66e3+2β68e1+2β72e42β73e7+2β74e62β75e12β77e3+2β78e2).

Now by applying D on e1=12[e2,e3], e1=12[e4,e5] and e1=12[e6,e7], we get D(e1)=12(2β32e2+2β33e12β35e7+2β36e62β37e5+2β38e42β42e3+2β44e1+2β45e6+2β46e72β47e42β48e5)=12(2β52e4+2β53e72β54e6+2β55e1+2β57e32β58e22β62e52β63e62β64e7+2β66e1+2β67e2+2β68e3)=12(2β72e62β73e52β74e4+2β75e3+2β76e22β77e1+2β82e7+2β83e42β84e52β85e2+2β86e32β88e1). Now by applying D on [e1,e3], [e4,e6] and [e5,e7], we get D(e2)=12(2β22e2+2β23e12β25e7+2β26e62β27e5+2β28e4+2β43e32β44e2+2β45e52β46e42β47e7+2β48e6)=12(2β52e72β53e4+2β54e5+2β55e22β56e3+2β58e12β72e52β73e62β74e7+2β76e1+2β77e2+2β78e3)=12(2β62e62β63e52β64e4+2β65e3+2β66e22β67e1+2β82e42β83e7+2β84e62β85e12β87e3+2β88e2). Similarly by applying D on all the remaining identities, we get for D(e4) D(e4)=12(2β22e4+2β23e72β24e6+2β25e1+2β27e32β28e2+2β63e32β64e2+2β65e52β66e42β67e7+2β68e6)=12(2β32e72β33e4+2β34e5+2β35e22β36e3+2β38e12β72e3+2β74e1+2β75e6+2β76e72β77e42β78e5)=12(2β42e62β43e52β44e4+2β45e3+2β46e22β47e1+2β82e22β83e1+2β85e72β86e6+2β87e52β88e4). For D(e5), D(e5)=12(2β22e5+2β23e6+2β24e72β26e12β27e22β28e3+2β53e32β54e2+2β55e52β56e42β57e7+2β58e6)=12(2β32e62β33e52β34e4+2β35e3+2β36e22β37e12β82e3+2β84e1+2β85e6+2β86e72β87e42β88e5)=12(2β42e72β43e4+2β44e5+2β45e22β46e3+2β48e1+2β72e22β73e1+2β75e72β76e6+2β77e52β78e4). For D(e6), D(e6)=12(2β22e62β23e52β24e4+2β25e3+2β26e22β27e1+2β83e32β84e2+2β85e52β86e42β87e7+2β88e6)=12(2β32e5+2β33e6+2β34e72β36e12β37e22β38e32β52e3+2β54e1+2β55e6+2β56e72β57e42β58e5)=12(2β42e4+2β43e72β44e6+2β45e1+2β47e32β48e2+2β62e22β63e1+2β65e72β66e6+2β67e52β68e4). For D(e7), D(e7)=12(2β22e72β23e4+2β24e5+2β25e22β26e3+2β28e1+2β73e32β74e2+2β75e52β76e42β77e7+2β78e6)=12(2β32e4+2β33e72β34e6+2β35e1+2β37e32β38e22β62e3+2β64e1+2β65e6+2β66e72β67e42β68e5)=12(2β42e5+2β43e6+2β44e72β46e12β47e22β48e3+2β52e22β53e1+2β55e72β56e6+2β57e52β58e4). By comparing the coefficients, we get βi1=0 for 2i8, βij=0 for 2i,j8 with i=j and βji=βij for 2i,j8 with ij. Specifically, β45=β27β36, β46=β28+β35, β47=β25+β38, β48=β26β37, β67=β23+β58, β68=β24β57 and β78=β34+β56◻

Theorem 2. Let x=i=18xiei1O. Let D:OO be a Lie derivation. Then D(x) can be written as D(x)=x1β11e0+(x3β23x4β24x5β25x6β26x7β27x8β28)e1+(x2β23x4β34x5β35x6β36x7β37x8β38)e2+(x2β24+x3β34+x5(β27+β36)+x6(β28β35)+x7(β25β38)+x8(β26+β37))e3+(x2β25+x3β35+x4(β27β36)x6β56x7β57x8β58)e4+(x2β26+x3β36+x4(β28+β35)+x5β56+x7(β23β58)+x8(β24+β57))e5+(x2β27+x3β37+x4(β25+β38)+x5β57+x6(β23+β58)+x8(β34β56))e6+(x2β28+x3β38+x4(β26β37)+x5β58+x6(β24β57)+x7(β34+β56))e7.

Proof. Applying D on x gives D(x)=i=18xiD(ei1). Substituting the values of D(ei)s, which is computed in matrix representation of D in above theorem yields D(x)=x1β11e0+x2(β23e2+β24e3+β25e4+β26e5+β27e6+β28e7)+x3(β23e1+β34e3+β35e4+β36e5+β37e6+β38e7)+x4(β24e1β34e2+(β27β36)e4+(β28+β35)e5+(β25+β38)e6+(β26β37)e7)+x5(β25e1β35e2+(β27+β36)e3+β56e5+β57e6+β58e7)+x6(β26e1β36e2+(β28β35)e3β56e4+(β23+β58)e6+(β24β57)e7)+x7(β27e1β37e2+(β25β38)e3β57e4+(β23β58)e5+(β34+β56)e7)+x8(β28e1β38e2+(β26+β37)e3β58e4+(β24+β57)e5+(β34β56)e6). Summarizing the above expression yields our required result. ◻

4. Characterizing Lie Derivation of Octonion Algebra O

Our next task is to present characterization of Lie derivations of the algebra of Octonion. In Theorem 2.2 of [21], it is shown that if S be a 2-torsion free ring and R=H(S) be quaternion ring, then every Lie derivation of R can be decomposed in terms of Jordan derivation and Lie derivation of S and an inner derivation of R, for every element tR. Here, we have:

Theorem 3. Let D:OO be a Lie derivation. Then there exist an element A in O, a Lie derivation δ and a Jordan derivation ψ on L such that D(t)=δ(x1)e0+i=28ψ(xi)ei1+IA(t) for every element t=i=18xiei1O.

Proof. Since D is an additive map, we can write (2)D(ei1)=j=18βijej1,1i8. for some βijsL. It can be easily seen that β11Z(L). Next, we will fine D(sei)s with i=0,1,,7, for arbitrary lL. Set D(le1)=i=18xiei1. Applying D on [le1,e1], we get 0=D[le1,e1]=2x3e3+2x4e22x5e5+2x6e4+2x7e72x8e6+(lβ23)e3(lβ24)e2+(lβ25)e5(lβ26)e4(lβ27)e7+(lβ28)e6. By comparing the coefficients, we get x3=12(lβ23),x4=12(lβ24),x5=12(lβ25),x6=12(lβ26),x7=12(lβ27),x8=12(lβ28), which implies D(le1)=x1e0+x2e1+12(lβ23)e2+12(lβ24e3+12(lβ25)e4+12(lβ26)e5+12(lβ27)e6+12(lβ28)e7. Now, applying D on the identities le3=12[le1,e2], le2=12[le1,e3], le1=12[le2,e3], le4=12[le1,e5], le5=12[le1,e4], le6=12[le1,e7], le7=12[le1,e6] and putting x2=ψ(l) where ψ:LL is an additive map which is uniquely determined by D, we get D(le1)=12Iβ34(l)e0+ψ(l)e1+12(lβ23)e2+12(lβ24e3+12(lβ25)e4+12(lβ26)e5+12(lβ27)e6+12(lβ28)e7.D(le2)=12Iβ24(l)e012(lβ23)e1+ψ(l)e2+12(lβ34)e3+12(lβ35)e4+12(lβ36)e5+12(lβ37)e6+12(lβ38)e7.D(le3)=12Iβ23(l)e012(lβ24)e112(lβ34)e2+ψ(l)e3+12(l(β27β36))e4+12(l(β28+β35))e512(l(β25β38))e612(l(β26+β37))e7D(le4)=12Iβ26(l)e012(lβ25)e112(lβ25)e212(l(β27β36))e3(3)+ψ(l)e4+12(lβ56)e5+12(lβ57)e6+12(lβ58)e7D(le5)=12Iβ25(l)e012(lβ26)e112(lβ36)e212(l(β28+β35))e312(lβ56)e4+ψ(l)e5+12(l(β23+β58))e6+12(l(β24β57))D(le6)=12Iβ28(l)e012(lβ27)e112(lβ37)e2+12(l(β25β38))e312(lβ57)e412(l(β23+β58))e5+ψ(l)e6+12(l(β34+β56))D(le7)=12Iβ27(l)e012(lβ28)e112(lβ38)e2+12(l(β26+β37))e312(lβ58)e412(l(β24β57))e512(l(β34+β56))e6+ψ(l)e7. Next, let lL be arbitrary and put D(le0)=D(l)=x1e0+x2e1+x3e2+x4e3+x5e4+x6e5+x7e6+x8e7. Applying D on [le0,e1]=0 and using (2), we obtain 0=D[le0,e1]=2x3e3+2x4e22x5e4+2x6e4+2x7e72x8e6+Iβ23(l)e2+Iβ24(l)e3+Iβ25(l)e4+Iβ26(l)e5+Iβ27(l)e6+Iβ28(l)e7. By comparing the coefficients, we get x3=12Iβ24(l),x4=12Iβ23(l),x5=12Iβ26(l),x6=12Iβ25(l),x7=12Iβ28(l),x8=12Iβ27(l). Applying D on the identities [le0,ei] where i=2,,7 and taking x1=δ(l) for some additive map δ:LL uniquely determined by D, we get (4)D(le0)=δ(l)e012Iβ34(l)e1+12Iβ24(l)e212Iβ23(l)e3+12Iβ26(l)e412Iβ25(l)e512Iβ28(l)e6+12Iβ27(l)e7 and Iβ34=Iβ56=Iβ34+β56Iβ24=Iβ57=Iβ24β57(5)Iβ23=Iβ58=Iβ23+β58Iβ26=Iβ37=Iβ26+β37Iβ25=Iβ38=Iβ25+β38Iβ28=Iβ35=Iβ28+β35Iβ27=Iβ36=Iβ27β36. Replacing l by [l1,l2] in (4), for some l1,l2L, we infer that δ is a Lie derivation of L. Moreover, applying D on the identity [l1e1,l2e2]=(l1l2)e3 and using the foregoing calculations, we can see that ψ is a Jordan derivation. Now let t=i=18xiei1O be an arbitrary element. Using (???), (4) and (???), we find that D(t)=δ(x1)e0+i=28ψ(xi)ei1+h(t) where h(t)=e02(Iβ34(x2)Iβ24(x3)+Iβ23(x4)Iβ26(x5)+Iβ25(x6)+Iβ28(x7)Iβ27(x8))+e12(Iβ34(x1)(x3β23)(x4β24)(x5β25)(x6β26)(x7β27)(x8β28))+e22(Iβ24x1+(x2β23)(x4β34)(x5β35)(x6β36)(x7β37)(x8β38))+e32(Iβ23x1+(x2β24)+(x3β34)(x5(β27β36))(x6(β28+β35))+(x7(β25β38))+(x8(β26+β37)))+e42(Iβ26x1+(x2β25)+(x3β35)+(x4(β27β36))(x6β56)(x7β57)(x8β58))+e52(Iβ25(x1)+(x2β26)+(x3β36)+(x4(β28+β35))+(x5β56)(x7(β23+β58))(x8(β24β57)))+e62(Iβ28x1+(x2β27)+(x3β37)(x4(β25β38))+(x5β57)+(x6(β23+β58))(x8(β34+β56)))+e72(Iβ27x1+(x2β28)+(x3β38)(x4(β26+β37))+(x5β58)+(x6(β24β57))+(x7(β34+β56))). It can be easily verified that h(t)=IA(t) where t=i=18xiei1 and A=12(β34e1+β24e2β23e3+β26e4β25e5β28e6+β27e7). Consequently, D(t)=δ(x1)e0+i=28ψ(xi)ei1+IA(t). ◻

5. Lie Centralizer of Octonion Algebra O

This section contains the characterization of Lie centralizer of octonion algebra. In Theorem 2.1 of [2], it is shown that if S is a 2-torsion free unital ring and R=H(S) is quaternion ring. Then every Lie centralizer of R can be represented in terms of a Lie centralizer and Jordan centralizer of S. Here, we have:

Theorem 4. Let ξ:OO be a Lie centralizer. Then there exists a Lie centralizer α and a Jordan centralizer φ on L such that ξ(t)=α(x1)e0+i=28φ(xi)ei1 for every element t=i=18xiei1O.

Proof. We have already assumed the form ξ(ei1)=j=18βijej1,1i8 for some βijsL. Since ξ is a Lie centralizer, we have ξ(e3)=12ξ[e1,e2]=12[ξ(e1),e2]=12(2β22e32β24e12β25e62β26e7+2β27e4+2β28e5)=β24e1+β22e3+β27e4+β28e5β25e6β26e7. Furthermore, ξ(e1)=12ξ[e2,e3]=12[ξ(e2),e3]=β33e1β32e2+β38e4β37e5+β36e6β35e7. By comparing the coefficients, we have β21=β24=β42=β41=β43=0, β22=β33=β44, β25=β38=β47, β26=β37=β48, β27=β36=β45, β28=β35=β46, which reduces ξ(e1) and ξ(e3) to ξ(e1)=β22e1+β23e2+β25e4+β26e5+β27e6+β28e7ξ(e3)=β22e3+β27e4+β28e5β25e6β26e7. Applying ξ on the identities e2=12[e3,e1], e4=12[e5,e1], e5=12[e1,e4], e6=12[e1,e7], e7=12[e6,e1], we get ξ(e2)=β22e2+β28e4β27e5+β26e6β25e7ξ(e4)=β28e2+β27e3+β22e4+β23e7ξ(e5)=β26e1β27e2+β28e3+β22e5+β23e6ξ(e6)=β27e1+β26e2+β25e3β23e5+β22e6ξ(e7)=β25e2β26e3β23e4+β22e7. Now assume that, ξ(e0)=t=i=18xiei1. We have 0=ξ[e0,t]=te0e0t=2x3e3+2x4e22x5e5+2x6e4+2x7e72x8e6, which implies x3=x4=x5=x6=x7=x8=0. Application of ξ on the identity [e0,e2] gives x2=0, which implies ξ(e0)=x1e0=x1L. Let sL be an arbitrary then 0=ξ[e0,se1]=[ξ(e0),se1]=(x1ssx1). From this we get, x1=ξ(e0)Z(L).
Let lL and set ξ(le1)=i=18xiei1. Applying ξ on [le1,e1]=0, we get x3=x4=x5=x6=x7=x8=0, which reduces ξ(le1) to ξ(le1)=x1e0+x2e1. Now applying ξ on the identities le3=12[le1,e2], le2=12[le3,e1], le1=12[le2,e3], le5=12[le1,e4], le4=12[le5,e1], le6=12[le1,e7], le7=12[le6,e1] and taking x2=φ(l), where φ:LL is an additive map which is uniquely determined by ξ, we get ξ(le1)=φ(l)e1,ξ(le2)=φ(l)e2,ξ(le3)=φ(l)e3,(6)ξ(le4)=φ(l)e4,ξ(le5)=φ(l)e5,ξ(le6)=φ(l)e6,ξ(le7)=φ(l)e7 Our next goal is to calculate ξ(le0) for arbitrary lL. Set ξ(l)=i=18xiei1. Applying ξ on [l,e1]=0 and [l,e2]=0 and putting x1=α(l), where α:LL is an additive map which is uniquely determined by ξ, we get (7)ξ(l)=α(l). Since ξ is a Lie centralizer, (7) implies that α is a Lie centralizer on L. Let l1,l2L. Applying ξ on the identity [l1e1,l2e2]=(l1l2)e3 and using (???) and (7), we get φ(l1l2)=φ(l1)l2 shows that φ is a Jordan centralizer on L. Now let t=i=18xiei1 be an arbitrary element in L. By (???) and (7), we get ξ(t)=α(x1)e0+i=28φ(xi)ei1, which completes the proof. ◻

6. Generalized Lie Derivation of Octonion Algebra O

Generalized derivation is an extension of natural derivation. It has many applications in the literature since it is quite helpful in the geometric classification of rings and algebras. In this section, we compute the matrix representation of generalized Lie derivation of the octonion algebra. Let F:OO be a generalized Lie derivation. F admits a matrix representation with respect to the basis, which is an 8×8 matrix [F]=(γij)T whose entries are defined by (8)F(ei1)=j=18γijej1,1i8. Each column of [F] is an element of O.

Theorem 5. Let F:OO be a generalized Lie derivation of O and B be the basis of O. Then the matrix representation of F is as follows =(γ11β1100000000γ2200000000γ2200000000γ2200000000γ2200000000γ2200000000γ2200000000γ22)+(β11000000000β23β24β25β26β27β280β230β34β35β36β37β380β24β340β27+β36β28β35β25β38β26+β370β25β35β27β360β56β57β580β26β36β28+β35β560β23β58β24+β570β27β37β25+β38β57β23+β580β34β560β28β38β26β37β58β24β57β34+β560).

Proof. Let F be a generalized Lie derivation of O, then (9)F[ei,ej]=[F(ei),ej]+[ei,D(ej)] where D is the derivation of the octonion algebra.
Put i=0, then F[e0,ej]=[F(e0),ej]+[e0,D(ej)]=0 for 1j7. So, by using the equation(8), we get γ1j=0 for 2j7.
Unlike the procedure of finding the Lie derivation of O, we don’t need to verify equation (9) for the products between the octonionic units to compute the matrix representation of F. Hence generalized Lie derivation is much easier to compute once a Lie derivation is obtained.
Suppose that F(ei1)=j=18γijej1,1i8. Applying F on the identity [ei,ej] for i=j and comparing the coefficients, we get γij=βij for 2i,j8 with ij.

By using the same technique proposed in the previous theorem, we get, for F(e1) F(e1)=12(2γ32e2+2γ33e12γ35e7+2γ36e62γ37e5+2γ38e42β42e3+2β44e1+2β45e6+2β46e72β47e42β48e5)=12(2γ52e4+2γ53e72γ54e6+2γ55e1+2γ57e32γ58e22β62e52β63e62β64e7+2β66e1+2β67e2+2β68e3)=12(2γ72e62γ73e52γ74e4+2γ75e3+2γ76e22γ77e1+2β82e7+2β83e42β84e52β85e2+2β86e32β88e1). For F(e2) F(e2)=12(2γ22e2+2γ23e12γ25e7+2γ26e62γ27e5+2γ28e4+2β43e32β44e2+2β45e52β46e42β47e7+2β48e6)=12(2γ52e72γ53e4+2γ54e5+2γ55e22γ56e3+2γ58e12β72e52β73e62β74e7+2β76e1+2β77e2+2β78e3)=12(2γ62e62γ63e52γ64e4+2γ65e3+2γ66e22γ67e1+2β82e42β83e7+2β84e62β85e12β87e3+2β88e2). For F(e3) F(e3)=12(2γ22e32γ24e12γ25e62γ26e7+2γ27e4+2γ28e5+2β33e32β34e2+2β35e52β36e42β37e7+2β38e6)=12(2γ52e62γ53e52γ54e4+2γ55e3+2γ56e22γ57e12β82e52β83e62β84e7+2β86e1+2β87e2+2β88e3)=12(2γ62e72γ63e4+2γ64e5+2γ65e22γ66e3+2γ68e1+2β72e42β73e7+2β74e62β75e12β77e3+2β78e2). For F(e4) F(e4)=12(2γ22e4+2γ23e72γ24e6+2γ25e1+2γ27e32γ28e2+2β63e32β64e2+2β65e52β66e42β67e7+2β68e6)=12(2γ32e72γ33e4+2γ34e5+2γ35e22γ36e3+2γ38e12β72e3+2β74e1+2β75e6+2β76e72β77e42β78e5)=12(2γ42e62γ43e52γ44e4+2γ45e3+2γ46e22γ47e1+2β82e22β83e1+2β85e72β86e6+2β87e52β88e4). For F(e5) F(e5)=12(2γ22e5+2γ23e6+2γ24e72γ26e12γ27e22γ28e3+2β53e32β54e2+2β55e52β56e42β57e7+2β58e6)=12(2γ32e62γ33e52γ34e4+2γ35e3+2γ36e22γ37e12β82e3+2β84e1+2β85e6+2β86e72β87e42β88e5)=12(2γ42e72γ43e4+2γ44e5+2γ45e22γ46e3+2γ48e1+2β72e22β73e1+2β75e72β76e6+2β77e52β78e4). For F(e6) F(e6)=12(2γ22e62γ23e52γ24e4+2γ25e3+2γ26e22γ27e1+2β83e32β84e2+2β85e52β86e42β87e7+2β88e6)=12(2γ32e5+2γ33e6+2γ34e72γ36e12γ37e22γ38e32β52e3+2β54e1+2β55e6+2β56e72β57e42β58e5)=12(2γ42e4+2γ43e72γ44e6+2γ45e1+2γ47e32γ48e2+2β62e22β63e1+2β65e72β66e6+2β67e52β68e4). For F(e7) F(e7)=12(2γ22e72γ23e4+2γ24e5+2γ25e22γ26e3+2γ28e1+2β73e32β74e2+2β75e52β76e42β77e7+2β78e6)=12(2γ32e4+2γ33e72γ34e6+2γ35e1+2γ37e32γ38e22β62e3+2β64e1+2β65e6+2β66e72β67e42β68e5)=12(2γ42e5+2γ43e6+2γ44e72γ46e12γ47e22γ48e3+2β52e22β53e1+2β55e72β56e6+2β57e52β58e4). By comparing the coefficients, we get γi1=0 for 2i8 and γ22=γ33=γ44=γ55=γ66=γ77=γ88◻

Theorem 6. Let x=x1e0+x2e1+x3e2+x4e3+x5e4+x6e5+x7e6+x8e7O. Let F:OO be a generalized Lie derivation with respect to D then F(x) can be written as F(x)=x1γ11e0+(x2γ22x3β23x4β24x5β25x6β26x7β27x8β28)e1+(x2β23+x3γ22x4β34x5β35x6β36x7β37x8β38)e2+(x2β24+x3β34+x4γ22+x5(β27+β36)+x6(β28β35)+x7(β25β38)+x8(β26+β37))e3+(x2β25+x3β35+x4(β27β36)+x5γ22x6β56x7β57x8β58)e4+(x2β26+x3β36+x4(β28+β35)+x5β56+x6γ22+x7(β23β58)+x8(β24+β57))e5+(x2β27+x3β37+x4(β25+β38)+x5β57+x6(β23+β58)+x7γ22+x8(β34β56))e6+(x2β28+x3β38+x4(β26β37)+x5β58+x6(β24β57)+x7(β34+β56)+x8γ22)e7.

Proof. Applying F on x gives F(x)=x1F(e0)+x2F(e1)+x3F(e2)+x4F(e3)+x5F(e4)+x6F(e5)+x7F(e6)+x8F(e7). Substituting the values of F(ei)s, which is computed in matrix representation of F in above theorem yields F(x)=x1γ11e0+x2(γ22e1+β23e2+β24e3+β25e4+β26e5+β27e6+β28e7)+x3(β23e1+γ22e2+β34e3+β35e4+β36e5+β37e6+β38e7)+x4(β24e1β34e2+γ22e3+(β27β36)e4+(β28+β35)e5+(β25+β38)e6+(β26β37)e7)+x5(β25e1β35e2+(β27+β36)e3+γ22e4+β56e5+β57e6+β58e7)+x6(β26e1β36e2+(β28β35)e3β56e4+γ22e5+(β23+β58)e6+(β24β57)e7)+x7(β27e1β37e2+(β25β38)e3β57e4+(β23β58)e5+γ22e6+(β34+β56)e7)+x8(β28e1β38e2+(β26+β37)e3β58e4+(β24+β57)e5+(β34β56)e6+γ22e7). Summarizing the above expression yields our required result. ◻

Example 1. Let an arbitrary element x=i=18xiei1O. Let β23=1, β26=1, β37=1, β58=1 and βij=0 otherwise in Lie derivation of O then D will be (10)D(x)=(x3x6)e1+(x2x7)e2+2x8e3x8e4+(x22x7)e5+(x3+2x6)e6+(2x4+x5)e7. Select γ11=1 and γ22=1 in the generalized Lie derivation, then (11)F(x)=x1e0+(x2x3x6)e1+(x2+x3x7)e2+(x4+2x8)e3+(x5x8)e4+(x2+x62x7)e5+(x3+2x6+x7)e6+(2x4+x5+x8)e7. Put i=1, j=7 in F[ei,ej]=[F(ei),ej]+[ei,D(ej)], we get F[e1,e7]=[F(e1),e7]+[e1,D(e7)]. The left hand side of the above equation will be F[e1,e7]=2F(e6)=2e24e5+2e6. By using (11), we get F(e1)=e1+e2+e5 and using (10), we get D(e7)=2e3e4.
Then by direct calculation the right hand side will be 2e64e52e2.

7. Conclusion

Lie algebras of derivations and its variants explore the nature of given algebras. In this research article, we have described of matrix representation as well as the characterization of Lie derivation of Octonion algebra. The characterization of Lie centralizer of Octonion algebra is also presented. We have also computed the matrix representation of generalized Lie derivations of Octonion algebras.

Acknowledgements

This research has been supported by HEC of Pakistan under the project NRPU 13150. Authors are thankful to HEC of Pakistan for supporting and funding this research.

Conflict of Interest

The authors declare no conflict of interests.

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